34 problems

Fomin–Zelevinsky's conjectures. The following statements hold:

Let $t_0\xrightarrow{k}t_1$ be adjacent vertices of $\mathbb{T}_n$, let $B^1=\mu_k(B^0)$, and fix $t\in\mathbb{T}_n$ and $\mathbf{a}\in\mathbb{Z}_{\geq0}^n$. Write…

Let $\mathcal{A}$ be a cluster algebra with initial seed at $t_0$, and let $\mathbf{g}_{\mathbf{a};t}^{B^0;t_0}$ be the $\mathbf{g}$-vector of a cluster monomial $x_{\mathbf{a};t}$…

Fix $B^0,t_0,\ell,t$ such that $F_{\ell;t}^{B^0;t_0}\neq1$, and write $\mathbf{g}_{\ell;t}^{B^0;t_0}=(g_1,\dots,g_n)$. Let $\operatorname{Trop}(u_1,\dots,u_n)$ denote the tropical…

Let $t_0\xrightarrow{k}t_1$ be adjacent vertices with $B^1=\mu_k(B^0)$, and let $h_k,h'_k$ and $g_k$ be the integers defined from the relevant $F$-polynomial specializations and th…

Let $\mathcal{Q}^{\leq \xi}_l$ be the principal quiver of $\Gamma^{\leq \xi}_l$, let $\mathcal{C}^{\leq \xi}_l$ be its associated cluster category, and let…

Reading's separation conjecture. The variables $u$ and $u'$ are contained in the same cluster if and only if they are sign-coherent.

Let $\Lambda$ be a finite-dimensional algebra and let $g$ be a g-vector. Write $[g]_{\operatorname{TF}}$ for the $\operatorname{TF}$-equivalence class of $g$, let…

Let $\Lambda$ be a finite-dimensional algebra, let $K_0(\operatorname{proj}\Lambda)_{\mathbb{R}}$ be its real Grothendieck group, and let $g$ be a g-vector. Write…

Let $\Lambda$ be the finite-dimensional algebra under consideration, and let … be a generically $\tau$-reduced component, with $|\Lambda|$ denoting the number of isomorphism classe…

Let $F_{i;v}$ be the $F$-polynomial associated with the cluster variable $x_{i;v}$, and let $\mathbf{g}(x_{i;v})=(g_1,\dots,g_m)$ be its g-vector. Write $u_1,\dots,u_m$ for tropica…

Positive dual c-vector uniqueness conjecture. Any positive dual $\boldsymbol{c}$-vector supports exactly one extremal exchangeable pair.

Let $d$ be a positive integer. For a cubical $d$-polytope $Q$, let $g^c(Q)$ be its cubical $g$-vector, and let $\mathcal{C}_d$ be the minimal closed cone containing all such vector…

The polytopal cellulation conjecture. There exists a unique polytopal complex $\mathcal C$ in $\mathbb R^d$ such that: (i) one face of $\mathcal C$ is the cross-polytope…

Let $P$ be a centrally symmetric simplicial $d$-polytope. The higher cs lower bound conjecture. If, for some $3\leq r<\left\lfloor\frac d2\right\rfloor$, … then … Here…

Fix a regular tree $\mathbb T_n$. For vertices $t_1,t_2\in\mathbb T_n$ joined by an edge labeled $k$, write $t_1\overset{k}{--}t_2$, let $B^2=\mu_k(B^1)$, and for $a\in[1,n]$ and…

Let $\mathcal A(\Sigma)$ be a cluster algebra with principal coefficients at $\Sigma=(X,Y,B)$, where $B$ is an $n\times n$ sign-skew-symmetric integer matrix. For a cluster $X'$ an…

Let $\mathbb{T}_n$ be the $n$-regular tree indexing seeds, let $t_0\stackrel{k}{\text{---\!---}}t_1$ be an edge, and let $B_0$ and $B_1$ be exchange matrices with $B_1=\mu_k(B_0)$.…

Let $B$ be an exchange matrix and let $t_1,\ldots,t_m$ be vertices of the $n$-regular tree $\mathbb T_n$. Let $i_1,\ldots,i_m\in[n]$ and let $c_1,\ldots,c_m\in\mathbb Z$. For every…

Let $B$ be an exchange matrix with principal coefficients and initial vertex $t_0$. Let $x_{i;t'}^{B;t_0}$ and $x_{j;t”}^{B;t_0}$ be cluster variables. For any vertex $t$ of the…

Let $B$ be an exchange matrix, let $t_0$ be a fixed initial vertex of the $n$-regular tree $\text{T}_n$, and let $t$ be any vertex. The vectors $\boldsymbol{g}_{i;t}^{B;t_0}$ are t…

Let $\text{T}_n$ be the $n$-regular tree indexing seeds, let $t_0$ and $t_1$ be adjacent vertices in direction $k$, and let $B_0$ and $B_1$ be exchange matrices with…

Assume $B$ is acyclic. Let $\nu:V\to V^*$ and $\eta:V^*\to V$ be the mutually inverse maps defined from the bilinear forms $E$ and $F$ in the source. For a cluster variable $x$, le…

A cluster monomial is a monomial in cluster variables from one seed, its support is the set of variables with nonzero exponent, and its $\mathbf g$-vector is the product of the…

For each vertex $v$ of the exchange graph, let $\operatorname{Cone}(v)$ be the cone in $V^*$ spanned by the weights $\mathbf g(x^v_e)$ for $e\in I(v)$. A fan is a collection of con…